The Given Line Segment Has A Midpoint At ( − 1 , − 2 (-1,-2 ( − 1 , − 2 ].What Is The Equation, In Slope-intercept Form, Of The Perpendicular Bisector Of The Given Line Segment?A. Y = − 4 X − 4 Y = -4x - 4 Y = − 4 X − 4 B. Y = − 4 X − 6 Y = -4x - 6 Y = − 4 X − 6 C. Y = 1 4 X − 4 Y = \frac{1}{4}x - 4 Y = 4 1 ​ X − 4 D.

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Introduction

In geometry, the perpendicular bisector of a line segment is a line that passes through the midpoint of the segment and is perpendicular to the segment. This concept is crucial in various mathematical applications, including coordinate geometry and trigonometry. In this article, we will explore the equation of the perpendicular bisector of a given line segment, with a midpoint at (−1,−2)(-1,-2).

Understanding the Midpoint

The midpoint of a line segment is the point that divides the segment into two equal parts. In the given problem, the midpoint is located at (−1,−2)(-1,-2). This means that if we were to draw a line segment with endpoints at two arbitrary points, the midpoint would be the point that lies exactly in the middle of the segment.

Slope-Intercept Form

The slope-intercept form of a line is a mathematical representation of a line in the form y=mx+by = mx + b, where mm is the slope of the line and bb is the y-intercept. The slope of a line is a measure of how steep the line is, while the y-intercept is the point at which the line intersects the y-axis.

Finding the Slope of the Perpendicular Bisector

To find the equation of the perpendicular bisector, we need to determine the slope of the line. Since the perpendicular bisector is perpendicular to the line segment, its slope will be the negative reciprocal of the slope of the line segment.

Calculating the Slope of the Line Segment

Let's assume that the endpoints of the line segment are (x1,y1)(x_1,y_1) and (x2,y2)(x_2,y_2). The slope of the line segment can be calculated using the formula:

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

However, since we are given the midpoint of the line segment, we can use the midpoint formula to find the coordinates of the endpoints:

(x1,y1)=(2(−1)+x2,2(−2)+y2)(x_1,y_1) = (2(-1) + x_2, 2(-2) + y_2)

(x2,y2)=(2(−1)−x2,2(−2)−y2)(x_2,y_2) = (2(-1) - x_2, 2(-2) - y_2)

Substituting these values into the slope formula, we get:

m=2(−2)+y2−(2(−2)+y2)2(−1)+x2−(2(−1)−x2)m = \frac{2(-2) + y_2 - (2(-2) + y_2)}{2(-1) + x_2 - (2(-1) - x_2)}

Simplifying the expression, we get:

m=00m = \frac{0}{0}

This indicates that the slope of the line segment is undefined, which means that the line segment is vertical.

Finding the Slope of the Perpendicular Bisector

Since the perpendicular bisector is perpendicular to the line segment, its slope will be the negative reciprocal of the slope of the line segment. However, since the slope of the line segment is undefined, the slope of the perpendicular bisector will be zero.

Finding the Equation of the Perpendicular Bisector

Now that we have the slope of the perpendicular bisector, we can find its equation using the slope-intercept form. Since the slope is zero, the equation of the perpendicular bisector will be of the form y=by = b, where bb is the y-intercept.

Finding the Y-Intercept of the Perpendicular Bisector

To find the y-intercept of the perpendicular bisector, we can use the fact that the perpendicular bisector passes through the midpoint of the line segment. Since the midpoint is located at (−1,−2)(-1,-2), we can substitute these values into the equation of the perpendicular bisector to find the y-intercept:

y=by = b

−2=b-2 = b

Therefore, the y-intercept of the perpendicular bisector is −2-2.

Conclusion

In conclusion, the equation of the perpendicular bisector of a line segment with a midpoint at (−1,−2)(-1,-2) is y=−2y = -2. This equation represents a horizontal line that passes through the midpoint of the line segment.

Final Answer

The final answer is y=−2y = -2.

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Introduction

In our previous article, we explored the equation of the perpendicular bisector of a line segment with a midpoint at (−1,−2)(-1,-2). In this article, we will answer some frequently asked questions related to the topic.

Q: What is the perpendicular bisector of a line segment?

A: The perpendicular bisector of a line segment is a line that passes through the midpoint of the segment and is perpendicular to the segment.

Q: How do I find the equation of the perpendicular bisector?

A: To find the equation of the perpendicular bisector, you need to determine the slope of the line. Since the perpendicular bisector is perpendicular to the line segment, its slope will be the negative reciprocal of the slope of the line segment. If the slope of the line segment is undefined, the slope of the perpendicular bisector will be zero.

Q: What if the line segment is vertical?

A: If the line segment is vertical, the slope of the line segment is undefined, and the slope of the perpendicular bisector will be zero. In this case, the equation of the perpendicular bisector will be of the form y=by = b, where bb is the y-intercept.

Q: How do I find the y-intercept of the perpendicular bisector?

A: To find the y-intercept of the perpendicular bisector, you can use the fact that the perpendicular bisector passes through the midpoint of the line segment. Substitute the coordinates of the midpoint into the equation of the perpendicular bisector to find the y-intercept.

Q: What if the line segment has a midpoint at (x,y)(x,y)?

A: If the line segment has a midpoint at (x,y)(x,y), the equation of the perpendicular bisector will be of the form y=by = b, where bb is the y-intercept. To find the y-intercept, substitute the coordinates of the midpoint into the equation of the perpendicular bisector.

Q: Can the perpendicular bisector be a horizontal or vertical line?

A: Yes, the perpendicular bisector can be a horizontal or vertical line. If the line segment is vertical, the perpendicular bisector will be a horizontal line. If the line segment is horizontal, the perpendicular bisector will be a vertical line.

Q: How do I graph the perpendicular bisector?

A: To graph the perpendicular bisector, use the equation of the line and plot the points on a coordinate plane. If the line is horizontal, plot a horizontal line through the midpoint of the line segment. If the line is vertical, plot a vertical line through the midpoint of the line segment.

Q: What is the significance of the perpendicular bisector?

A: The perpendicular bisector is significant in geometry and trigonometry. It is used to find the midpoint of a line segment, the equation of a line, and the slope of a line. It is also used in various real-world applications, such as architecture, engineering, and computer graphics.

Conclusion

In conclusion, the perpendicular bisector of a line segment is a line that passes through the midpoint of the segment and is perpendicular to the segment. The equation of the perpendicular bisector can be found using the slope of the line segment and the coordinates of the midpoint. The perpendicular bisector is significant in geometry and trigonometry and has various real-world applications.

Final Answer

The final answer is that the equation of the perpendicular bisector of a line segment with a midpoint at (−1,−2)(-1,-2) is y=−2y = -2.